Leaf holonomy of Reeb foliation on Möbius strip

Question

I am trying to understand the leaf holonomy of the Reeb foliation on the Möbius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am currently reading Hector and Hirsch's "Introduction to the geometry of foliations" Part A and in the book they describe Reeb components of the annulus to asymptotically approach the closed boundary leaves. However, the Möbius strip obviously only has one boundary component, so this is difficult transfer the visual and understand the holonomic behavior in this case.

The way that I have been thinking about it is in terms of the holonomy either "attracting" or "repelling" from the circle leaf on the boundary. Does anyone have any suggestions for understanding or even writing down the leaf holonomy of the Reeb foliation on the Möbius strip, or any helpful ways to visualize this? Additionally, it would be helpful to see what the quotient space representation looks like with the foliation.

Answer 1

A good reading! If you are working in the smooth ($C^\infty$) regularity class, then the Reeb foliations $F$ on the Möbius strip $M$ are in infinite number, up to diffeomorphism. Fix a small enough compact arc $A=[a,b]$ transverse to $F$ and with one endpoint $a$ on the boundary. Then, for one of the two orientations of the boundary, the first return map $r:A\to A$ is contracting, in the sense that it is a diffeomorphism of $[a,b]$ onto $[a, r(b)]$, with $a<r(b)<b$, fixing $a$ and such that $r(x)<x$ for every $x\in(a,b]$. Conversely, every germ at $a$ of such a contraction can be realized by a Reeb foliation of the Möbius strip. Beware that, although every leaf of $F$ meets $A$, the space of leaves is NOT the quotient of $A$ by $r$: because we are on the Möbius strip rather than the usual strip, there is moreover, in the holonomy of the foliation on $A$, a diffeomorphism $s:(a,b]\to(a,s(b)]$, with $a<s(b)<b$, such that $s\circ s=r$. You can realize $s$ by going across the strip along the leaves, rather than staying close to the boundary. You will see $s$ if you make a paper model and draw a Reeb component on it. The space of leaves is the quotient topological space $A/R$ where $x R y$ iff there is an $n\in Z$ such that $y=s^n(x)$: this is a circle plus one point adherent to all points of the circle. You can also see this by considering the circle $C$ which is in the middle of $M$: it meets transversely at one point every leaf, but the boundary leaf.